Flat Plate Collector (Physical)

from watertap_contrib.reflo.solar_models import FlatPlatePhysical

Note

This model was designed for use in a model utilizing the multiperiod framework. Steady-state applications are not recommended.

This Flat Plate Collector (FPC) unit model is a physical model that inherits its base model structure from the Solar Energy Base Class. A flat plate collector (FPC) converts solar energy into thermal energy that is transferred to a heat transfer fluid. Solar collectors usually operate with low and variable energy and do not track the sun, thus requiring minimal maintenance. They can deliver moderate temperatures up to 100°C. The following model uses physical equations to predict the useful heat generated by an FPC as a function of its design, heat transfer fluid and irradiance.

Degrees of Freedom & Variables

The model has 6 degrees of freedom that should be fixed for the unit to be fully specified. The following 4 variables define the input stream state variables.

Variables	Variable Name	Symbol	Units
Inlet mass flow rate liquid water	`flow_mass_phase_comp['Liq','H2O']`	\(m_{l}\)	\(\text{kg} / \text{s}\)
Inlet mass flow rate vapor water	`flow_mass_phase_comp['Vap','H2O']`	\(m_{v}\)	\(\text{kg} / \text{s}\)
Inlet temperature	`temperature`	\(T_{f}\)	\(\text{K}\)
Inlet pressure	`pressure`	\(P_{in}\)	\(\text{Pa}\)

The following variables must be fixed by the user for a fully defined model.

Variables	Variable Name	Symbol	Units
Collector area	`collector_area`	\(A_{c}\)	\(\text{m}^2\)
Total irradiance	`total_irradiance`	\(G_{total}\)	\(\text{W}/\text{m}^2\)

Model Structure

This flat plate collector model consists of 2 StateBlocks assigned to the inlet and outlet ports of the heating fluid.

inlet_block
outlet_block

The mass flow rate of liquid and vapor water, temperature, and pressure at the inlet port must be specified by the user.

Sets

The model consists of the phase set included in the property package.

Description	Symbol	Indices
Time	\(t\)	[0]
Phases	\(p\)	[‘Liq’, ‘Vap’]

Parameters

The following parameters are used and are mutable (except for the test conditions).

Description	Parameter Name	Symbol	Value	Units
Number of collectors	`number_collectors`	\(n_{c}\)	1	\(\text{dimensionless}\)
Product of cover transmittance and shortwave absorptivity of absorber	`trans_absorb_prod`	\(\tau\alpha\)	1	\(\text{dimensionless}\)
Product of collector heat removal factor, cover transmittance, and shortwave absorptivity of absorber	`FR_ta`	\({F}_{R}\tau\alpha\)	0.689	\(\text{dimensionless}\)
Product of collector heat removal factor and overall heat loss coeff. of collector	`FR_UL`	\({F}_{R}{U}_{L}\)	3.85	\(\text{W}\text{/m}^2\text{/K}\)
Mass flow rate of fluid during characterization test (fixed)	`mdot_test`	\(\dot{m}_{test}\)	1	\(\text{kg/s}\)
Specific heat capacity of fluid during characterization test (fixed)	`cp_test`	\(c_{ptest}\)	4184	\(\text{J/kg/K}\)
Specific heat capacity of fluid being used for heat transfer in operation	`cp_use`	\(c_{use}\)	4184	\(\text{J/kg/K}\)
Pump power	`pump_power`	\(P_{pump}\)	1	\(\text{W}\)
Pump efficiency	`pump_eff`	\(\eta_{pump}\)	1	\(\text{dimensionless}\)
Ambient temperature	`temperature_ambient`	\(T_{amb}\)	303.15	\(\text{K}\)
Maximum irradiance at the location	`max_irradiance`	\(G_{max}\)	1000	\(\text{W}/\text{m}^2\)
Influent minus ambient temperature	`factor_delta_T`	\(\Delta T\)	0.03	\(\text{K}\)

Equations

The following equations calculate the variables used in estimating heat transfer in a flat plate collector.

Description	Variable Name	Equation	Units
Product of collector efficiency factor and overall heat loss coefficient at test conditions	`Fprime_UL`	\(F^{'}U_{L} = -(\dot{m}_{test}{c}_{ptest})/A_{c} log(1-{F}_{R}{U}_{L}A_{c}/(\dot{m}_{test}{c}_{ptest}))\)	\(\text{dimensionless}\)
Ratio of FRta_use to FRta_test	`ratio_FRta`	\(r = [m_{l}{c}_{ptest}/A_{c}][1 - \text{exp}(-A_{c}F^{'}U_{L}/m_{l}{c}_{ptest})]/F_{R}U_{L}\|_{test}\)	\(\text{dimensionless}\)
Useful net heat gain	`net_heat_gain`	\(P_{gain} = {n}_{c}A_{c}r(F_{R}\tau\alphaG_{total}\tau\alpha - F_{R}U_{L}(T_{f}-{T}_{amb}))\)	\(\text{W}\)
Rated plant heat capacity	`system_capacity`	\(P_{th} = {n}_{c}A_{c}(F_{R}\tau\alphaG_{total}\tau\alpha - F_{R}U_{L}*\Delta T )\)	\(\text{MW}\)
Annual thermal energy produced by the system	`heat_annual`	\(H_{annual} = P_{gain} * 8760\)	\(\text{kWh}\)

Costing

The costing approach is adopted from the SAM costing for flat plate collector systems. The following parameters are constructed on the costing block for FPC costing:

Cost Component	Variable	Symbol	Value	Units	Description
Cost per area collector	`cost_per_area_collector`	\(c_{c}\)	600	\(\text{USD/m}^2\)	Cost per area for solar collector
Cost per volume storage	`cost_per_volume_storage`	\(c_{hs}\)	120	\(\text{USD}\text{/m}^3\)	Cost per volume for thermal storage
Contingency factor	`contingency_frac_direct_cost`	\(X_{c}\)	0.07	\(\text{dimensionless}\)	Fraction of direct costs for contingency
Indirect cost factor	`indirect_frac_direct_cost`	\(X_{i}\)	0.11	\(\text{dimensionless}\)	Fraction of direct costs for indirect costs
Sales tax as fraction of capital costs	`sales_tax_frac`	\(X_{t}\)	0	\(\text{dimensionless}\)	Sales tax as fraction of capital costs
Fixed operating cost per system capacity	`fixed_operating_by_capacity`	\(c_{fix,op}\)	16	\(\text{USD/kW/year}\)	Fixed operating cost of flat plate plant per kW capacity

Cost Component	Symbol	Equation
Collector cost	\(C_{coll}\)	\(c_{c} \times A_{total}\)
Land Cost	\(C_{land}\)	\(c_{land} \times A_{land}\)
Fixed Operating Cost	\(C_{fix,op}\)	\(c_{fix,op} \times P_{th}\)

The direct costs include the cost of the collectors and contingency.

\[C_{direct} = C_{coll} * (1 + X_{c})\]

Indirect costs are calculated as a fraction of the direct costs and the land cost:

\[C_{indirect} = A_{land} c_{land} + C_{direct} X_{i}\]

The total capital cost of the FPC system is the sum of direct and indirect costs and sales tax:

\[C_{capital} = (C_{indirect} + C_{direct}) (1 + X_{t})\]

Note that by default, REFLO assumes no sales tax (i.e., \(X_{t} = 0\)) or land cost (i.e., \(c_{land} = 0\)).

The total operating cost is the fixed operating cost:

\[C_{operating} = C_{fix,op}\]

Energy Balance

The FPC model has both thermal and electric power flows. The steady-state thermal output of the FPC system is calculated as:

\[Q_{out} = H_{annual} / 8760\]

\(Q_{out}\) is the steady-state thermal output (in kW) at the target temperature
\(H_{annual}\) is the annual thermal energy generation (in kWh)

The parasitic power consumption of the FPC system is calculated as:

\[P_{cons} = E_{annual} / 8760\]

\(P_{cons}\) is the parasitic power consumption (in kW)
\(E_{annual}\) is the annual electric energy consumption (in kWh)

References

[1] Solar Engineering of Thermal Processes, Duffie and Beckman, 4th ed.